NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION

SARAJEVO JOURNAL OF MATHEMATICS Vol1 (5), No, (016), 05 15 DOI: 105644/SJM107 NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION WENCHANG CHU AND NADIA N LI Abstract By meas of two ad three term relatos of 3φ -seres, we vestgate the otermatg 4φ 3-seres Eght trasformato formulaes to double seres are establshed Four of them are show to be otermatg extesos of the Sears trasformato 1 Itroducto ad motvato For the two determates x ad, the shfted factoral of x wth base reads as (x; ) 0 1 ad (x; ) (1 x) (1 x) (1 1 x) for N Whe < 1, we have two well defed fte products (x; ) (1 x) ad (x; ) (x; ) / ( x; ) 0 The product ad fracto of shfted factorals are abbrevated respectvely to α, β,, γ; (α; ) (β; ) (γ; ), α, β,, γ (α; ) (β; ) (γ; ) A, B,, C (A; ) (B; ) (C; ) Followg Gasper Rahma 4, the basc hypergeometrc seres s defed by a0, a 1,, a r ; { 1+rφ s z ( 1) b 1,, b ( ) } s r a 0, a 1,, a r z s, b 1,, b s 0 where the base wll be restrcted to < 1 for otermatg -seres Most of the -seres results cocer the case r s Whe the parameters 010 Mathematcs Subject Classfcato Prmary 33D15; Secodary 05A15 Key words ad phrases Basc hypergeometrc seres; The -Pfaff Saalschütz theorem; The Sears trasformato; The -Kampé de Féret seres Copyrght c 016 by ANUBIH

06 WENCHANG CHU AND NADIA N LI satsfy the codto a 0 a 1 a a r b 1 b b r, the seres s sad to be balaced There are umerous mportat summato ad trasformato formulae the -seres theory Oe of them s due to Sears 5 (cf 4, III 15), whch cocers the termatg balaced 4 φ 3 -seres subject to the codto αβγ 1 bcd: 4φ, b, c, d α/c, γ/c 3 ; α, β, γ α, γ 4φ, c, β/b, β/d 3 ; (1) β, αβ/bd, βγ/bd The am of ths paper s to vestgate ts otermatg form b, c, d, e 4φ 3 ; αβγ ( c, e, bd/β αβγ ) b, d () α, β, γ bcde, α, γ 0 bcde β, bd/β Recall the -Pfaff Saalschütz summato theorem (cf 1, 84 ad 4, II-1) 3φ, a, b c/a, c/b c, 1 ; (3) ab/c c, c/ab Ths ca be utlzed to rewrte the afore-dsplayed 4 φ 3 -seres as b, c, d, e 4φ 3 ; αβγ ( c, e, bd/β αβγ α, β, γ bcde, α, γ 0 bcde ( c, e, bd/β αβγ ), α, γ 0 bcde ) 3φ, β/b, β/d β, 1 β/bd 0, β/b, β/d β, 1 β/bd ; Lettg + the last double sum ad the terchagg the summato order, we get the followg expresso b, c, d, e 4φ 3 ; αβγ ( c, e, β/b, β/d αγ ) α, β, γ bcde, α, β, γ 0 ce 3 φ c, e, bd/β α, ; αβγ (4) γ bcde Ths wll be our startg pot for the subseuet developmet By meas of the -Kummer Thomae Whpple trasformato ad the Hall trasformato (cf Gasper Rahma 4, III 9 & 10), we shall derve four double seres expressos for the 4 φ 3 -seres gve () the ext secto The the thrd secto, further four trasformato formulae wll be establshed by utlzg the two three term trasformatos (cf Gasper Rahma 4, III 33 & 34) Fally, the paper wll ed wth a dscusso of some ow results that are cotaed as partcular cases of the ma theorems proved ths paper

NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION 07 Trasformatos to double seres Based o the double sum expresso dsplayed (4), we shall derve four trasformato theorems ths secto The ma tools are the -Kummer Thomae Whpple trasformato (cf Gasper Rahma 4, III 9) 3φ b, c, d α, γ ; αγ bcd α/d, αγ/bc α, αγ/bcd 3φ d, γ/b, γ/c γ, αγ/bc ; α d ad the Hall trasformato (cf Gasper Rahma 4, III 10) b, c, d 3φ ; αγ b, αγ/bd, αγ/bc α/b, γ/b, αγ/bcd α, γ bcd α, γ, αγ/bcd 3φ ; b αγ/bd, αγ/bc (6) 1 Accordg to (5), we ca reformulate the 3 φ -seres dsplayed (4) as 3φ c, e, bd/β α, ; αβγ αγ/ce, αβ/bd γ bcde α, αβγ/bcde bd/β, γ/c, γ/e 3 φ αγ/ce, ; αβ γ bd Substtutg the last expresso to (4), we get the frst trasformato formula Theorem 1 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ αβ/bd, αγ/ce α, β, γ bcde α, αβγ/bcde ( c, e, β/b, β/d αγ ) bd/β, γ/c, γ/e (γ; ) +, β, αβ/bd ce, αγ/ce, 0 ( αβ ) bd Whe d 1, the last theorem gves, uder the replacemet b bβ, the followg closed formula for the double seres, whch s euvalet to Gasper s oe (cf Gasper Rahma 4, P300) Corollary (Summato formula of double seres) α, αγ/bce ( c, e, 1/b αγ ) ( b, γ/c, γ/e α ) α/b, αγ/ce (γ; ) +, α/b, 0 ce, αγ/ce b The last formula may be cosdered as a double seres exteso of the well ow -Gauss summato theorem (cf Baley1, 84 ad Gasper Rahma 4, II 8) because for γ e, t reduces euvaletly to the followg a, b φ 1 ; c c/a, c/b where c/ab < 1 (7) c ab c, c/ab (5)

08 WENCHANG CHU AND NADIA N LI Alteratvely, applyg (5) to the 3 φ -seres dsplayed (4) 3φ c, e, bd/β α, ; αβγ c, αγ/ce, αβγ/bcd γ bcde α, γ, αβγ/bcde α/c, γ/c, αβγ/bcde 3 φ αγ/ce, ; c αβγ/bcd we derve from (4) the secod trasformato formula Theorem 3 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ c, αγ/ce, αβγ/bcd α, β, γ bcde α, γ, αβγ/bcde c ( e, β/b, β/d αγ ) α/c, γ/c, αβγ/bcde (αβγ/bcd; ) +, β ce, αγ/ce, 0 Lettg β e bd the last theorem, we get the followg terestg reducto formula for the -Appell le Φ (3) -seres (cf Gasper Rahma 4, 10) Corollary 4 (Reducto formula of double seres) b, c, d 3φ ; αγ α, γ α, γ bcd c, αγ/c c b, d (αγ/c; ) +, 0 ( αγ bcd ) α/c, γ/c Ths detty exteds aga the -Gauss summato theorem (7), whch correspods, fact, to the partcular case d 1 of the formula dsplayed the last corollary 3 By meas of (6), the 3 φ -seres (4) ca be reformulated as 3φ c, e, bd/β α, ; αβγ γ bcde α/e, αβγ/bcd α, αβγ/bcde 3φ e, γ/c, βγ/bd γ, ; α αβγ/bcd e Substtutg ths expresso to (4), we get the thrd trasformato formula Theorem 5 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ α/e, αβγ/bcd α, β, γ bcde α, αβγ/bcde

NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION 09 e, βγ/bd γ, αβγ/bcd, 0 + ( αγ ce ) c, β/b, β/d, β, βγ/bd (γ/c; ) ( α ) (; ) e Lettg β 0 the last theorem, we fd aother terestg summato formula for the -Appell Φ (1) -seres (cf Gasper Rahma 4, 10) Corollary 6 (Summato formula of double seres) (α; ) (e; ) + (c; ) ( αγ ) (γ/c; ) ( α ) (α/e; ) (γ; ) + (; ) ce (; ) e, 0 Whe γ 0, ths detty recovers the -bomal seres (cf 1, 8 ad 4, II 3) a 1φ 0 ; x (ax; ) where x < 1 (8) (x; ) 4 Aalogously, by utlzg (6), we ca restate 3 φ -seres (4) as 3φ c, e, bd/β α, ; αβγ bd/β, αβγ/bcd, αβγ/bde γ bcde α, γ, αβγ/bcd αβγ/bcde, 3 φ αβ/bd, βγ/bd αβγ/bcd, ; bd αβγ/bde β Ths leads (4) to the fourth trasformato formula Theorem 7 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ bd/β, αβγ/bcd, αβγ/bde αβ bd, βγ bd α, β, γ bcde α, γ, αβγ/bcde αβγ, 0 bcd, αβγ bde c, e, β/b, β/d ( αγ ) (αβγ/bcde; ) ( bd ), β, αβ bd, βγ bd ce (; ) β Lettg d 1 ad the replacg b by bβ the last theorem, we obta the followg terestg summato formula Corollary 8 (Summato formula of double seres) α, γ, αγ/bce α/b, γ/b b, αγ/bc, αγ/be αγ/bc, αγ/be, 0 + c, e, 1/b (αγ ) (αγ/bce; ) b, α/b, γ/b ce (; ) Whe c 1, ths detty recovers aga the -Gauss summato theorem (7) +

10 WENCHANG CHU AND NADIA N LI 3 Two terms trasformatos I ths secto, four trasformato theorems volvg two double sums wll be establshed by employg the followg two three term trasformato formulae (cf Gasper Rahma 4, III 33 & 34): b, c, d 3φ ; αγ α/b, α/c b, c, γ/d α, γ abc α, α/bc 3φ ; γ, bc/α (9) b, c, γ/d, αγ/bc α/b, α/c, αγ/bcd + 3φ ; ; α, γ, bc/α, αγ/bcd b, c, d 3φ ; αγ α, γ bcd α/b, α/c, c/d, /γ α, c/γ, /d, α/bc /γ, α/γ, b, c, γ/d, αγ/bc, bc/αγ γ/, α, b/γ, c/γ, /d, α/bc, bc/α αγ/bc, α/bc c, γ/d, c/α 3φ ; b c/d, bc/α γ b γ 3φ, c γ, d γ γ, α ; αγ bcd γ (10) 31 Accordg to (9), we ca reformulate the 3 φ -seres (4) as 3φ c, e, bd/β α, ; αβγ α/e, αβ/bd γ bcde α, αβ/bde 3φ e, γ/c, bd/β ; γ, bde/αβ γ/c, bd/β, + e, αβγ/bde α/e, α, γ, bde/αβ, αβγ/bcde 3φ αβ/bd, αβγ/bcde ; αβγ/bde, αβ/bde Substtutg ths to (4) results the frst trasformato formula Theorem 9 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ α/e, αβ/bd (e; ) ( + αγ ) α, β, γ bcde α, αβ/bde (γ; ) + ce, 0 c, β/b, β/d γ/c, bd/β, β, αβ/bd, bde/αβ e, γ/c, bd/β, αβγ/bde + α, γ, bde/αβ, αβγ/bcde (αβ/bd; ) ( + c, β/b, β/d αγ ) α/e, αβγ/bcde (αβγ/bde; ) +, β, αβ/bd ce, αβ/bde, 0 Whe d 1, the last theorem yelds the followg two-term relato Corollary 10 (Two term summato formula) (αβγ/bce; ) α/e, αβ/b, αβγ/bce (αβγ/be; ) α, αβ/be, αβγ/be (e; ) ( + αγ ) c, β/b γ/c, b/β (γ; ) + ce, αβ/b, be/αβ e, γ/c, b/β + α, γ, be/αβ, 0

, 0 NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION 11 (αβ/b; ) + (αβγ/be; ) + c, β/b, αβ/b ( αγ ce ) α/e, αβγ/bce, αβ/be 3 By meas of (9), the 3 φ -seres dsplayed (4) ca be expressed as 3φ c, e, bd/β α, ; αβγ α/c, α/e γ bcde α, α/ce 3 φ c, e, βγ/bd 1+ ce/α, ; + c, e, βγ/bd, αγ/ce γ α, γ, ce/α, αβγ/bcde α/c, α/e, αβγ/bcde 3 φ αγ/ce, 1 ; α/ce The correspodg (4) yelds the secod trasformato formula Theorem 11 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ α/c, α/e α, β, γ bcde α, α/ce β/b, β/d, β, βγ/bd, 0 ( )+ ( γ) (α/ce; ), 0 c, e, βγ/bd ce/α, γ ()+ ( γ) c, e, αγ/ce, βγ/bd + (; ) α, γ, ce/α, αβγ/bcde β/b, β/d α/c, α/e, αβγ/bcde, β, βγ/bd, αγ/ce Whe d 1, the last theorem gves aother two-term relato Corollary 1 (Two term summato formula) (αβγ/bce; ) (αγ/ce; ) α/c, α/e, αβγ/bce α, α/ce, αγ/ce c, e, βγ/b + α, γ, ce/α, 0, 0 ( )+ ( γ) (α/ce; ) c, e, βγ/b ce/α, γ + β/b, βγ/b β/b, βγ/b + ( )+ ( γ) (; ) α/c, α/e, αβγ/bce, αγ/ce 33 Applyg (10) to the 3 φ -seres (4), we get the expresso 3φ c, e, bd/β α, ; αβγ γ 3 φ e, αβ/bd, e/γ bcde 1+ eβ/bd, 1+ ; c ce/γ α γ/c, γ/e, 1 /α, 1+ eβ/bd c/α, e/α, e/α, β/bd, γ, γ/ce 3 φ 1 bd/αβ γ/α, ; αβγ /α bcde

1 WENCHANG CHU AND NADIA N LI γ/α, 1 /α, c, e, αβ/bd, αγ/ce, ce/αγ 1 α, γ, c/α, e/α, β/bd, γ/ce, 1+ ce/γ Substtutg the last euato to (4) gves rse to the thrd trasformato formula Theorem 13 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ /α, γ/c, γ/e, βe/bd α, β, γ bcde γ, γ/ce, e/α, β/bd e, αβ/bd c, β/b, β/d (e/γ; ) eβ/bd, ce/γ, β, αβ/bd, 0 + (; ) + /α, γ/α, c, e, αβ/bd, αγ/ce, ce/αγ α α, γ, c/α, e/α, β/bd, γ/ce, ce/γ (α/; ) β/b, β/d c/α, e/α (αβ/bd; ), β, γ/α, 0 ( c ) α ( αγ ) ce For ths theorem ad the ext oe, we shall ot produce the two term relatos correspodg to the case d 1, whch wll resemble Corollares 10 ad 1 34 Fally, the 3 φ -seres dsplayed (4) ca be reformulated through (10) as 3φ c, e, bd/β α, ; αβγ γ/e, βγ/bd, 1 /α, 1 bd/cβ γ bcde βγ/bde, 1 /c, γ, 1 bd/αβ bd/β, α/c, 3 φ 1 bd/βγ bde/βγ, 1 ; e c bd/cβ 3 φ α, e α, 1 bd αβ α γ/α, ; αβγ /α bcde γ/α, 1 /α, e, bd/β, α/c, 1 αβγ/bde, bde/αβγ 1 α, γ, e/α, βγ/bde, 1 bd/αβ, 1 /c, bde/βγ whch leads (4) to the fourth trasformato formula Theorem 14 (Double sum expresso) b, c, d, e 4φ 3 ; αβγ γ/e, βγ/bd, bd/cβ, /α α, β, γ bcde γ, bd/αβ, /c, βγ/bde (cβ/bd; ) ( e, β/b, β/d αγ ) α/c, bd/β (βγ/bd; ), β, αβ/bd, 0 ce, bde/βγ + /α, γ/α, e, bd/β, α/c, αβγ/bde, bde/αβγ α α, γ, bd/αβ, e/α, /c, βγ/bde, bde/βγ ( ce ) αγ

NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION 13, 0 (α/; ) (αβ/bd; ) β/b, β/d, β c/α, e/α, γ/α ( αγ ) ce It s wdely ow that some two term relatos ca be rewrtte compact form terms of -tegrals For the -double seres, there s also such a example Gasper Rahma 4, Exercse 1016) However, t s ulely to do so for the theorems derved ths secto because the two double sums volved are very dfferet ther structure The oly excepto s Theorem 9 wth the two double sums havg the same symmetrc form However, t seems a ueasy tas to express t -tegrals due to the presece of varable (αγ/ce) 4 Further ow examples The four theorems establshed the last secto are all reduced to the Sears trasformato (1) whe the 4 φ 3 -seres s termatg balaced oe I fact, for αβγ bcde, the 4 φ 3 -seres Theorem 9 s balaced The correspodg result ca be stated as the followg proposto Proposto 15 (αβγ bcde) b, c, d, e α/e, αβ/bd 4φ 3 ; α, β, γ α, αβ/bde c, β/b, β/d (bd/β; ), β, αβ/bd c (; ) γ ( αγ ) c, β/b, β/d ce, β, αβ/bd, 0, 0 (e; ) ( + αγ (γ; ) + ce e, c, bd/β, α, γ ) (αβ/bd; ) + (α/e; ) (c; ) + (αβ/bde; ) +1 Whe e, the secod part wll be ahlated by the zero factor ( ; ), whle the frst sum ca be mapulated as follows ( ; ) ( + αγ ) c, β/b, β/d (bd/β; ) (γ; ) + c, β, αβ/bd +, 0 (; ) (, c, β/b, β/d αγ ), β, γ, αβ/bd φ, bd/β 1 0 c ; γ Evaluatg the φ 1 -seres by meas of the -Chu Vadermode Gauss formula (cf 4, II 6) φ, bd/β 1 ; ( βγ/bd; ) ( bd ) γ ( γ; ) β (βγ/bd; ) (γ; ) (βγ/bd; ) (γ; ) ( bd β )

14 WENCHANG CHU AND NADIA N LI we fd that the last double sum reduces to a sgle oe The the resultg expresso gves the Sears trasformato (1) after some route smplfcato Two further partcular cases of Proposto 15 may be worthy of metog The frst case α e results a specal form of (5) whe the seres s balaced Aother oe s the otermatg -Vadermode sum (cf 4, II 3) whch correspods to the case c 1 Corollary 16 (αβγ bde) a, b /c, ab/c φ 1 ; c a/c, b/c + /c, a, b a/c, b/c c c, a/c, b/c φ 1 ; /c Aalogously, we may exame Theorem 13 Whe αβγ bcde, the 4 φ 3 - seres s balaced The the frst double sum ca be reduced to a sgle oe e, αβ/bd (e/γ; ) ( c, β/b, β/d c ) eβ/bd, ce/γ, 0 + (; ), β, αβ/bd α c, e, β/b, β/d e/γ,, β, αβ/bd, eβ/bd φ e 1 1+ ; c eβ/bd 0 α c, e, β/b, β/d βγ/bd, β/bd, β, αβ/bd, βe/bd 1+ eβ/bd, c/α 0 βγ/bd, β/bd c, e, β/b, β/d eβ/bd, c/α 4φ 3 ;, β, αβ/bd, βγ/bd where the last φ 1 -seres has bee evaluated through (7) Hece, Theorem 13 uder the balaced codto αβγ bcde becomes the followg trasformato formula Proposto 17 (αβγ bcde) b, c, d, e /α, γ/c, γ/e, ce/α c, e, β/b, β/d 4φ 3 ; α, β, γ γ, c/α, e/α, γ/ce 4φ 3 ; β, αβ/bd, βγ/bd + /α, c, e, bd/β, γ/α α α, c/α, e/α, bd/αβ, γ β/b, β/d (α/) ( c/α, e/α bd ), β (αβ/bd), γ/α β, 0 Furthermore, we ca chec wthout dffculty that the termatg case e of ths proposto recovers aga the Sears trasformato (1) Istead, whe β d, Proposto 17 reduces to the followg otermatg -Pfaff-Saalschütz theorem (cf 4, II 4)

NONTERMINATING EXTENSIONS OF THE SEARS TRANSFORMATION 15 Corollary 18 (αγ bcd) b, c, d /γ, α/b, α/c, α/d 3φ ; α, γ α, b/γ, c/γ, d/γ + α/γ, /γ, b, c, d γ α, γ, b/γ, c/γ, d/γ 3φ b/γ, c/γ, d/γ /γ, α/γ ; Before edg the paper, we would le to pot out that all the theorems obtaed ths paper volve the double -seres Oe mportat class of them s called -Kampé de Féret Seres (cf Gasper Rahma 4, 10 for example) The formed reader ca chec wthout dffculty that Theorem 3 s, fact, euvalet to a trasformato due to Chu Ja, Proposto 3, whle Theorem 7 s essetally a reducto formula Further summato ad trasformato formulae ca be foud the papers by Chu et al, 3 Acowledgemet The authors are scerely grateful to a aoymous referee for the crtcal commets ad valuable suggestos, that helped us substatally to mprove the mauscrpt revso Refereces 1 W N Baley, Geeralzed Hypergeometrc Seres, Cambrdge Uversty Press, Cambrdge, 1935 W Chu ad C Ja, Trasformato ad reducto formulae for double -Clause hypergeometrc seres, Math Methods Appl Sc, 31 (1) (008), 1 17 3 W Chu ad N N L, Termatg -Kampé de Féret seres Φ 1:3;λ 1:;µ ad Φ:;λ :1;µ, Hroshma Math J, 4 () (01), 33 5 4 G Gasper ad M Rahma, Basc Hypergeometrc Seres (d ed), Cambrdge Uversty Press, Cambrdge, 004 5 D B Sears, O the trasformato theory of basc hypergeometrc fuctos, Proc Lodo Math Soc, 53 (1951), 158 180 (Receved: August 30, 015) (Revsed: Jauary 1, 016) Wechag Chu Dpartmeto d Matematca e Fsca Eo De Gorg Uverstá del Saleto Lecce Aresao P O Box 193 73100 Lecce Italy chuwechag@usaletot Nada N L Departmet of Mathematcs Zhouou Normal Uversty Zhouou 466000 P R Cha la3718@163com